Page 299 - Bird R.B. Transport phenomena

§9.7 Convective Transport of Energy 283

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For gas-filled granular beds ' a different type of complication arises. Since the thermal
conductivities of gases are much lower than those of solids, most of the gas-phase heat
conduction is concentrated near the points of contact of adjacent solid particles. As a re-
sult, the distances over which the heat is conducted through the gas may approach the
mean free path of the gas molecules. When this is true, the conditions for the develop-
ments of §9.3 are violated, and the thermal conductivity of the gas decreases. Very effec-
tive insulators can thus be prepared from partially evacuated beds of fine powders.
Cylindrical ducts filled with granular materials through which a fluid is flowing (in the z di-
rection) are of considerable importance in separation processes and chemical reactors. In
such systems the effective thermal conductivities in the radial and axial directions are
10
quite different and are designated by K eff//r and K efffZZ . Conduction, convection, and radia-
tion all contribute to the flow of heat through the porous medium. 11 For highly turbulent
flow, the energy is transported primarily by the tortuous flow of the fluid in the inter-
stices of the granular material; this gives rise to a highly anisotropic thermal conductivity.
For a bed of uniform spheres, the radial and axial components are approximately

K (9.6-8, 9)
eii,rr — K e ff,zz ~~
in which v is the "superficial velocity" defined in §4.3 and §6.4, and D is the diameter of
p
0
the spherical particles. These simplified relations hold for Re = D v p/fi greater than
p o
200. The behavior at lower Reynolds numbers is discussed in several references. 12 Also,
the behavior of the effective thermal conductivity tensor as a function of the Peclet num-
ber has been studied in considerable detail. 13
19.7 CONVECTIVE TRANSPORT OF ENERGY
In §9.1 we gave Fourier's law of heat conduction, which accounts for the energy trans-
ported through a medium by virtue of the molecular motions.
Energy may also be transported by the bulk motion of the fluid. In Fig. 9.7-1 we
show three mutually perpendicular elements of area dS at the point P, where the fluid

2,
V

X
P<
dS

Fig. 9.7-1. Three mutually perpendicular surface elements of area dS across which energy is
being transported by convection by the fluid moving with the velocity v. The volume rate of
flow across the faceperpendicular to the x-axis is v dS, and the rate of flow of energy across
x
2
dS is then ({pv + pU)v dS. Similar expressions can be written for the surface elements per-
x
pendicular to the y- and z-axes.
10
See Eq. 9.1-7 for the modification of Fourier's law for anisotropic materials. The subscripts rr and
zz emphasize that these quantities are components of a second-order symmetrical tensor.
"W. B. Argo and J. M. Smith, Chem. Engr. Progress, 49,443^51 (1953).
J. Beek, Adv. Chem. Engr., 3, 203-271 (1962); H. Kramers and K. R. Westerterp, Elements of Chemical
12
Reactor Design and Operation, Academic Press, New York (1963), §111.9; O. Levenspiel and К. В. Bischoff,
Adv. Chem. Engr., 4, 95-198 (1963).
D. L. Koch and J. F. Brady, /. Fluid Mech., 154,399-427 (1985).
13